Paul has some very encouraging thoughts about this in 1 Corinthians:
To the rest I say this (I, not the Lord): If any brother has a wife who is not a believer and she is willing to live with him, he must not divorce her. And if a woman has a husband who is not a believer and he is willing to live with her, she must not divorce him. For the unbelieving husband has been sanctified through his wife, and the unbelieving wife has been sanctified through her believing husband. Otherwise your children would be unclean, but as it is, they are holy.
Your prayers will have a great effect on the "atmosphere" in your home, and the enemy does have to obey you as you are united with Christ. We would suggest that you (privately) "renounce" the agreements and sins of your spouse first; plead the blood of Christ over them to help cancel the enemy's claims. Then invoke the very opposite things into your home and marriage and "upon" your spouse in the name of Jesus!
You may also find some of our prayers helpful, in staying centered with God and enforcing his Kingdom over your home.